Commensurability of hyperbolic manifolds with geodesic boundary

Suppose n >= 3, let M (1), M (2) be n-dimensional connected complete finite-volume hyperbolic manifolds with nonempty geodesic boundary, and suppose that pi(1) (M (1)) is quasi-isometric to pi(1) (M (2)) (with respect to the word metric). Also suppose that if n=3, then partial derivative M (1) and partial derivative M (2) are compact. We show that M (1) is commensurable with M (2). Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifolds with non-compact geodesic boundary which are not commensurable with each other. We also prove that if M is as M (1) above and G is a finitely generated group which is quasi-isometric to pi(1) (M), then there exists a hyperbolic manifold with geodesic boundary M' with the following properties: M' is commensurable with M, and G is a finite extension of a group which contains pi(1) (M') as a finite-index subgroup.